\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2019 (2019), No. 07, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2019 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2019/07\hfil Regularity criteria for weak solutions]
{Regularity criteria for weak solutions to the 3D Navier-Stokes
equations in bounded domains via BMO norm}

\author[J.-M. Kim \hfil EJDE-2019/07\hfilneg]
{Jae-Myoung Kim}

\address{Jae-Myoung Kim \newline
Center for Mathematical Analysis \& Computation,
Yonsei University,
Seoul, Korea}
\email{cauchy@naver.com}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted March 27, 2018. Published January 17, 2019.}
\subjclass[2010]{35Q30, 35B65, 30H35}
\keywords{Navier-Stokes equation; regularity criteria; BMO space}

\begin{abstract}
 We study three-dimensional incompressible Navier-Stokes
 equations in bounded domains with smooth boundary.
 We present  regularity criteria of weak solutions to this
 equation via the BMO norm.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

We study the three-dimensional Navier-Stokes equation
\begin{equation}\label{nse-10}
u_t+(u \cdot \nabla)u-\Delta{u}+\nabla\pi=0 ,\quad \operatorname{div} u=0
\quad \text{in } Q_T:=\Omega \times (0,T),
\end{equation}
where $\Omega$ is a domain in $\mathbb{R}^3$ with smooth boundary
$\partial \Omega \in C^2$. Here $u:Q_T\to \mathbb{R}^3$ is the flow
velocity vector and $ \pi:Q_T\to\mathbb{R}$ is the
pressure. We consider the initial-boundary value problem of
\eqref{nse-10} with  initial condition
\begin{equation}\label{nse-20}
u(x,0)=u_0(x) \quad \quad x\in\Omega
\end{equation}
together with two types of boundary conditions:
Either
\begin{equation}\label{noslip-slip}%(B1)\quad
u=0,
\end{equation}
or
\begin{equation}\label{double-slip} %(B2)
 u\cdot n=0, \quad (\nabla \times u)\times n=0,
\end{equation}
where $n$ is the outward unit normal vector along boundary
$\partial\Omega$. The initial conditions satisfy the compatibility
condition, i.e.\ $\nabla \cdot u_0(x)=0$. A weak solution $u$ of
\eqref{nse-10}--\eqref{nse-20} with boundary conditions either

\eqref{noslip-slip} or \eqref{double-slip} is regular in $Q_T$ provided that
$\|u\|_{L^{\infty}(Q_T)}<\infty$. The notion of weak solutions
will be introduced in Definition \ref{weak-solution} of Section 2.

The initial conditions hold the compatibility condition, i.e.\
$\nabla \cdot u_0(x)=0$. Since Leray \cite{Ler} proved the existence
of weak solutions of the Navier-Stokes equations (see also
\cite{Hopf}), regularity question has remained open.

\begin{definition} \rm
A weak solution $u$ of \eqref{nse-10}--\eqref{nse-20} with boundary conditions
\eqref{noslip-slip} or \eqref{double-slip} is regular in $Q_T$
provided that $\|u\|_{L^{\infty}(Q_T)}<\infty$.
\end{definition}

It is known that any weak solution becomes unique and regular in
$Q_T$, provided that the following scaling invariant conditions
\cite{Veiga95, CL02, Ser1, Z06}, so called Serrin's type conditions,
are satisfied:
\begin{gather*}
u \in L^{q}(0, T; L^{p}({\mathbb{R}}^3)),\quad  3/p+2/q \leq 1,
\; 3<p\leq \infty, \\
\nabla u \in L^{q}(0, T; L^{p}({\mathbb{R}}^3)), \quad  3/p+2/q
\leq 2,\; \frac{3}{2}<p\leq \infty, \\
\pi \in L^{q}(0, T; L^{p}({\mathbb{R}}^3)), \quad 3/p+2/q \leq
2,\; \frac{3}{2}<p\leq \infty, \\
\nabla \pi \in L^{q}(0, T; L^{p}({\mathbb{R}}^3)), \quad
 3/p+2/q \leq 3,\; 1<p\leq \infty.
\end{gather*}
In this direction, thee are numerous contributions,
see \cite{BCJ, ESS, FJR, G86, Lady, Ohyama, Prodi, Soh}. In view
of the regularity conditions in view of the BMO space, Kozono and
Taniuchi proved in \cite{KT} that a weak solution $u$ become regular
if $u$ satisfies
\begin{gather*}
u \in L^2(0,T ;\operatorname{BMO}( \mathbb{R}^3), \\
w:=\nabla\times u \in L^1(0,T ;\operatorname{BMO}( \mathbb{R}^3),\quad T<\infty,
\end{gather*}
which is the result to the  space BMO, which is larger than
$L^\infty (\mathbb{R}^3)$. Also, Fan and Ozawa proved in \cite{FO} that a
weak solution $u$ become regular if $u$ satisfies
\[
\nabla p \in L^{2/3}(0, T; \operatorname{BMO}(\mathbb{R}^3)),\quad 0<T<\infty.
\]
Our study is motivated by the works above, that is, we obtain the
regularity conditions for a weak solution to 3D Naiver-Stokes
equations \eqref{nse-10}--\eqref{nse-20} with the boundary
conditions \eqref{noslip-slip} or \eqref{double-slip} in bounded
domains. In particular, for bounded domains, the difficulty lies in
treating the pressure. To be more precise, in the case that
$\Omega=\mathbb{R}^3$, using the equation of pressure, we observe that the
pressure $\pi$ satisfies
\begin{equation}\label{cd}
\|\pi\|_{L^p(\mathbb{R}^n)}\leq C\|u\|^2_{L^{2p}(\mathbb{R}^3)},\quad
1<p<\infty.
\end{equation}
However, it is not known yet whether or not the estimate above
\eqref{cd} holds for domains with the boundary condition. Thus, the
methods of proof in a whole space $\mathbb{R}^3$ do not seem to be
applicable to our case. To overcome these difficulties, we use the
maximal estimates of Stokes system for both cases of slip and
no-slip boundary conditions, regarding the nonlinear term as an
external force (see Lemma \ref{lem1} in section 2). Since such
estimates of the Stokes system are also available for domain with
boundaries, this approach allows for control of pressure and is
useful for our analysis. On the other hand, to obtain the regularity
condition for a vorticity vector, we consider the vorticity
equations for Navier-Stokes equations to avoid the estimate of terms
containing the pressure term. In this case, our proof is based on a
priori estimate for the vorticity. At last, we give regularity
criteria for the pressure to this equations using the maximal
regularity theorem (see Lemma \ref{lem1} in section 2).
Our main results read as follows.


\begin{theorem}\label{thm1}
Suppose that $u$ is a weak solution to
\eqref{nse-10}--\eqref{nse-20} with initial conditions
$u_0\in H^2(\Omega)\cap W^{1,q}(\Omega)$, $q>3$ and boundary conditions
\eqref{noslip-slip} or \eqref{double-slip}. Assume further that $u$ satisfies
\begin{equation*} \label{thm1-mag}
\|u\|_{L^2(0,T;\operatorname{BMO}(\Omega))}<\infty
\end{equation*}
Then, $u$ becomes regular in $\overline{Q_T}$.
\end{theorem}

\begin{theorem}\label{thm2}
Suppose that $u$ is a weak solution to
\eqref{nse-10}--\eqref{nse-20} with initial conditions $u_0\in
H^2(\Omega)\cap W^{1,q}(\Omega)$, $q>3$ and boundary conditions
\eqref{noslip-slip} or \eqref{double-slip}.
Assume further that $w:=\nabla \times u$ satisfies
\begin{equation*} %\label{thm2-mag}
\|w\|_{L^{1}(0,T;\operatorname{BMO}(\Omega))}<\infty
\end{equation*}
Then, $u$ becomes regular in $\overline{Q_T}$.
\end{theorem}

\begin{theorem}\label{thm3}
Suppose that $u$ is a weak solution to
\eqref{nse-10}--\eqref{nse-20} with initial conditions $u_0\in
H^2(\Omega)\cap W^{1,q}(\Omega)$, $q>3$ and boundary condition
\eqref{noslip-slip}. Assume further that $u$ satisfies
\begin{equation*} \label{thm3-mag}
\|\pi\|_{L^2(0,T;\operatorname{BMO}(\Omega))}<\infty
\end{equation*}
Then, $u$ become regular in $\overline{Q_T}$.
\end{theorem}

\begin{theorem}\label{thm4}
Suppose that $u$ is a weak solution to
\eqref{nse-10}--\eqref{nse-20} with initial conditions $u_0\in
H^2(\Omega)\cap W^{1,q}(\Omega)$, $q>3$ and boundary conditions
\eqref{noslip-slip} or \eqref{double-slip}. Assume further that $u$ satisfies
\begin{equation*} \label{thm4-mag}
\|\nabla \pi\|_{L^{2/3}(0,T;\operatorname{BMO}(\Omega))}<\infty
\end{equation*}
Then, $u$ becomes regular in $\overline{Q_T}$.
\end{theorem}

\begin{remark} \rm
Theorem \ref{thm3} can be extended to any dimension $\Omega \subset
\mathbb{R}^N$, because we do not deal with the terms related to pressure. On
the other hand, Theorems \ref{thm1}, \ref{thm2} and \ref{thm4} can
be restricted to the case $n = 3, 4$ in view of \cite[Remark 3.2]{KL06}
or \cite{K10}.
\end{remark}

\begin{remark} \rm
 Theorems \ref{thm4} is given in \cite{FLN17} under the
boundary condition \eqref{double-slip}.  For the convenience of readers, we give
a sketch of the proof.
\end{remark}

This article is organized as follows. In Section 2, we recall the
notion of weak solutions and review some known results.
In Section 3, we present the proofs of Theorems \ref{thm1}--\ref{thm4}.


\section{Preliminaries}

 In this section, we introduce the notation and
definitions used throughout this paper. We also recall some lemmas
which are useful for our analysis.  For $1 \leq q \leq \infty $ and a
nonnegative integer $k$, $W^{k,q}( \Omega )$ indicates the standard
Sobolev space with norm $\|\cdot\|_{k,q}$, i.e.,
$W^{k,q}(\Omega) = \{ u \in L^{q}( \Omega ): D^{ \alpha }u \in L^{q}( \Omega ), 0
\leq | \alpha | \leq k \}$. As usual, $W^{k,q}_0(\Omega)$ is defined
as the completion of $\mathcal{C}^{\infty}_0(\Omega)$  in
$W^{k,q}(\Omega)$. When $q=2$, we write $W^{k,q}(\Omega)$ as
$H^{k}(\Omega)$. Let $I$ be a finite time interval. For a function
$f(x,t)$, $\Omega\subset\mathbb{R}^3$, we denote
$\|f\|_{L^{p,q}_{x,t}(\Omega\times I)}=\|f\|_{L^{q}_{t}(I;
L^p_x(\Omega))}=\big\|\|f\|_{L^p_x(\Omega)}\big\|_{L^q_t(I)}$. All
generic constants will be denoted by $C$,
which may vary from line to line.
We recall first the definition of weak solutions.

\begin{definition} \label{weak-solution} \rm
Let $u_0 \in L^2(\Omega)$ with $\operatorname{div} u_0 = 0$. We say that
$u$ is a distributional solution (or weak solution) of
\eqref{nse-10}--\eqref{nse-20} if $u$ satisfies the following:
\begin{itemize}
\item[(1)]  $u \in L^{\infty}(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega))$ and
$u$ satisfies
\begin{equation*}
\int_0^{T}\int_{\Omega}\Big(\frac{\partial\phi}{\partial
t}+(u\cdot\nabla)\phi\Big) u\,dx\,dt
+\int_{\Omega}u_0\phi(x,0)dx
=\int_0^{T}\int_{\Omega}\nabla u :\nabla\phi \,dx\,dt
\end{equation*}
for all $\phi\in
\mathcal{C}_0^{\infty}(\Omega\times [0,T))$ with
$\operatorname{div} \phi=0$.

\item[(2)] $u$ satisfies divergence free condition; that is,
$\int_{\Omega} u\cdot\nabla\psi dx=0$ for any
$\psi\in\mathcal{C}^{\infty}(\bar{\Omega})$.
\end{itemize}
\end{definition}

We consider the following Stokes system which is the linearized
Navier-Stokes equations,
\begin{equation}\label{stokes-eqn}
v_t-\Delta v+\nabla p=f, \quad  \operatorname{div}v=0 \quad \text{in }
Q_T:=\Omega\times (0,T)
\end{equation}
with initial data $v(x,0)=v_0(x)$. As in \eqref{noslip-slip} and
\eqref{double-slip}, boundary data of $v$ are again assumed to be
either no-slip or slip conditions, namely
\begin{gather}\label{stokes-boundayconditions1}
v(x,t)=0,\quad x\in\partial\Omega\quad\text{or } \\
\label{stokes-boundayconditions2}
v\cdot n=0, \quad   (\nabla \times v)\times n=0,\quad
x\in\partial\Omega.
\end{gather}
Next, we recall maximal estimates of the Stokes system in terms of
mixed norms (see \cite[Theorem 5.1]{GS} and \cite[Theorem 1.2]{Shi}
for no-slip and slip boundary cases, respectively).

\begin{lemma}\label{lem1}
Let $1<l,m<\infty$. Suppose that $f\in L^{l,m}_{x,t}(Q_T)$ and
$v_0\in D_l^{1-\frac{1}{m},m}$.
If $(v,p)$ is the solution of the Stokes system
\eqref{stokes-eqn} satisfying one of the boundary conditions
\eqref{stokes-boundayconditions1} or
\eqref{stokes-boundayconditions2}, then the following estimate is
satisfied:
\begin{equation}\label{stokes-estimate}
\begin{aligned}
&\|v_t\|_{L^{l,m}_{x,t}(Q_T)}+\|\nabla^2 v\|_{L^{l,m}_{x,t}(Q_T)}
+\|\nabla p\|_{L^{l,m}_{x,t}(Q_T)} \\
&\leq C\|f\|_{L^{l,m}_{x,t}(Q_T)}+\|v_0\|_{D_l^{1-\frac{1}{m},m}(\Omega)}\,.
\end{aligned}
\end{equation}
\end{lemma}
We note that $\|v_0\|_{D_l^{1-\frac{1}{m},m} (\Omega)}
\leq \|v_0\|_{W^{1,l}(\Omega)}$
because
\[
D_l^{1-\frac{1}{m},m}(\Omega):=[L_l(\Omega),W^{1,l}((\Omega))]_{1-\frac{1}{m},m}
\]
(see e.g., \cite[Chapter 7]{AF}) and, therefore,
$\|v_0\|_{D_l^{1-\frac{1}{m},m}(\Omega)}$ in
\eqref{stokes-estimate} can be replaced by
$\|v_0\|_{W^{1,l}(\Omega)}$.

The John-Nirenberg space or the space of the Bounded Mean
Oscillation (in short BMO space) \cite{JN} consists of all functions
$f$ which are integrable on every ball $B_R(x) \subset \mathbb{R}^3$ and
satisfy
$$
\|f\|^2_{\rm BMO}= \sup_{x\in \mathbb{R}^3} \sup_{R>0}
\frac{1}{B(x,R)}\int_{B(x,R)} | f (y)- f_{B_R} (y)|dy<\infty.
$$
Here, $f_{B_R}$ is the average of $f$ over all ball $B_R(x)$ in
$\mathbb{R}^3$.
Next we recall a Gagliardo-Nirenberg inequality using BMO-norm
(See \cite[Theorem 2.3]{DX} and \cite[Theorem 2.2]{KW}).

\begin{lemma}\label{esti-bmo}
Suppose that $1 \leq p < r < \infty$ and
$f \in L^p(\Omega)\cap \operatorname{BMO}(\Omega)$. Then there exists a constant
$C=C(n,p,r,\Omega)$ such
that
$$\|f\|_{L^r(\Omega)} \leq
C\|f\|_{L^p(\Omega)}^{p/r}\|f\|_{\operatorname{BMO}(\Omega)}^{1-\frac{p}{r}}.
$$
\end{lemma}

Also, we recall estimates with respect to smooth vector field under
the slip boundary condition.
(See \cite[Lemma 2.2]{Veiga09},\cite[Theorem 2.1]{Veiga10} and \cite[Lemma
2.1-2.2]{BS12}).

\begin{lemma}\label{gradient}
Let $\Omega$ be a smooth domain in $\mathbb{R}^3$. Then, for each $q > 1$,
regular smooth vector fields $f$,
\begin{itemize}
\item[(a)]
\begin{align*}
-\int_{\Omega}\Delta f \cdot f|f|^{q-2}dx
&=\frac{1}{2}\int_{\Omega}|f|^{q-2}|\nabla f|^2dx
+\frac{4(q-2)}{q^2}\int_{\Omega}\big|\big|f|^{q/2}\big|^2dx \\
&\quad -\int_{\partial \Omega}|f|^{q-2}(n\cdot \nabla )f \cdot fdS.
\end{align*}

\item[(b)] Moreover, using the vector identity,
\[
(n\cdot \nabla)f \cdot f=(f\cdot \nabla)f\cdot n+((\nabla \times f)
\times n)\cdot f,
\]
we  deduce that
\begin{align*}
&-\int_{\Omega}\Delta f \cdot f|f|^{q-2}dx \\
&=\frac{1}{2}\int_{\Omega}|f|^{q-2}|\nabla f|^2dx
+\frac{4(q-2)}{p^2}\int_{\Omega}\big|\nabla |f|^{q/2}\big|^2dx \\
&\quad -\int_{\partial \Omega}|f|^{p-2}(f\cdot \nabla )f \cdot
ndS-\int_{\partial \Omega}|f|^{p-2}((\nabla \times f)\times
n)\cdot fdS.
\end{align*}
\end{itemize}
\end{lemma}

\begin{lemma}\label{estimate-boudnary}
Assume that $u$ is a regular enough satisfying the boundary
condition \eqref{double-slip} on $\partial \Omega$. Then, the
for $w=\nabla \times u$ we have
\[
-\frac{\partial w}{\partial n}\cdot
w=(\epsilon_{1jk}\epsilon_{1\beta\gamma}+\epsilon_{2jk}
\epsilon_{2\beta\gamma}+\epsilon_{3jk}\epsilon_{3\beta\gamma})
w_jw_{\beta}\partial_kn_{\gamma} \ \text{on} \partial \Omega,
\]
where $\epsilon_{ijk}$ denotes the totally anti-symmetric tensor
such that $(a \times b)=\epsilon_{1jk}a_jb_k$. In particular,
\[
\int_{\Omega}\Delta w \cdot wdx\leq -\int_{\Omega}|\nabla w|^2dx
+C\int_{\partial \Omega}|w|^2 dx.
\]
\end{lemma}

\section{Proof of main results}


\begin{proof}[Proof of Theorem \ref{thm1}]
Following the argument in \cite{KK12, K10}, it is sufficient to show
the $L^4$-estimate of $u$. Suppose that $T^*$ be the first time of
singularity. Then $u$ must satisfies for any $\delta>0$,
\begin{equation}\label{1stpf-20}
\begin{aligned}
&\lim_{t\nearrow T^*}\Big(\|u(\cdot,t)\|^4_{L^4} +\int_{T^*
-\delta}^t \Big(\big\| |\nabla u(\cdot,\tau)|\, |u(\cdot,\tau)|
\big\|^2_{L^2} \\
&\quad +\big\|\nabla|u(\cdot,\tau)|^2 \big\|^2_{L^2}\Big)d\tau\Big)=\infty.
\end{aligned}
\end{equation}
In the proof, we consider only the  boundary condition
\eqref{double-slip}, since the case of \eqref{noslip-slip} is much simpler.
Multiplying the first equation of \eqref{nse-10} with $|u|^2u$, and integrating
over $\Omega$, we have
\begin{equation}\label{fluid-L4}
\begin{aligned}
&\frac{1}{4}\frac{d}{dt}\int_{\Omega}|u|^4
 +\int_{\Omega}|\nabla u|^2|u|^2
 +\frac{1}{2}\int_{\Omega}|\nabla|u|^2|^2 \\
&=-\int_{\Omega} \nabla \pi|u|^2u
+\sum_{i,j=1}^3\int_{\partial\Omega}u_{j,x_i} u_j|u|^2 n_i,
\end{aligned}
\end{equation}
where we used integration by parts, divergence-free conditions of
$u$ and trace theorem.
Let $\epsilon$ be a sufficiently small positive number, which will
be specified later. Integrating \eqref{fluid-L4} in time over
$(T^*-\epsilon, \tau)$ for any $\tau$ with $T^*-\epsilon<\tau<T^*$,
we observe that
\begin{equation}\label{fluid-inme-space}
\begin{aligned}
&\frac{1}{4}\int_{\Omega}|u(\cdot,\tau)|^4 dx
 -\frac{1}{4}\int_{\Omega}|u(\cdot, T^*-\epsilon)|^4 dx \\
&+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|\nabla u|^2|u|^2 \,dx\,dt
+\frac{1}{2}\int_{T^*-\epsilon}^{\tau}\int_{\Omega}\big|\nabla|u|^2\big|^2\,dx\,dt\\
&\leq \int_{T^*-\epsilon}^{\tau}\int_{\Omega} |\nabla \pi| |u|^2
|u|\,dx\,dt
+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|u|^3|\nabla u|\,dx\,dt:=I_1+I_2.
\end{aligned}
\end{equation}
For convenience, we denote $Q_{\tau}:=\Omega\times (T^*-\epsilon, \tau)$.
Using H\"older's inequality, the first term $I_1$ can be estimated as follows:
\begin{align*}
I_1
&\leq \int_{T^*-\epsilon}^{\tau}\|\nabla\pi\|_{L^2}
\|u\|^3_{L^{6}} \leq C\int_{T^*-\epsilon}^{\tau}
\|\nabla\pi\|_{L^2}\|u\|^2_{L^4} \|u\|_{\rm BMO} \\
&\leq C\|\nabla \pi\|_{L^2(Q_{\tau})}
\|u\|_{L^2(0,\tau;\operatorname{BMO}(\Omega))}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4},
\end{align*}
For convenience of computations, we denote
$\mathcal{C}_{\epsilon}:=\|u(\cdot,T^*-\epsilon)\|_{W^{1,2}(\Omega)}$.
Using the estimate \eqref{stokes-estimate}, we continue to estimate
$I_1$ as
\begin{align}
I_1
&\leq C\Big(\|(u \cdot \nabla) u\|_{L^2(Q_{\tau})}+\mathcal{C}_{\epsilon}\Big)\|u\|_{L^2((T^*-\epsilon,\tau);\operatorname{BMO}(\Omega))}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4} \nonumber \\
&\leq C\|(u \cdot \nabla) u\|_{L^2(Q_{\tau})}
 \|u\|_{L^2((T^*-\epsilon,\tau);\operatorname{BMO}(\Omega))}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4}
\label{estimate-200} \\
&\quad +C\mathcal{C}_{\epsilon}\|u\|_{L^2(0,\tau;\operatorname{BMO}(\Omega))}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4}. \nonumber
\end{align}
On the other hand, by direct calculations, $I_2$ is bounded by
$$
C\epsilon^{1/2}\big||u||\nabla u|\big|_{L^2(Q_{\tau})}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4}.
$$
Summing the estimates of $I_1$ and $I_2$ with using Young's inequality, we
obtain
\begin{align*}
&\frac{1}{4}\int_{\Omega}|u(\cdot,\tau)|^4 dx
 -\frac{1}{4}\int_{\Omega}|u(\cdot, T^*-\epsilon)|^4 dx \\
&+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|\nabla u|^2|u|^2 \,dx\,dt
+\frac{1}{2}\int_{T^*-\epsilon}^{\tau}\int_{\Omega}\big|\nabla|u|^2\big|^2\,dx\,dt \\
&\leq C\|\,|u| |\nabla u|\,\|_{L^2(Q_{\tau})}
\|u\|_{L^2(0,\tau;\operatorname{BMO}(\Omega))}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4} \\
&\quad +CC_{\epsilon}\|u\|_{L^2(0,\tau;\operatorname{BMO}(\Omega))}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4} \\
&\leq \frac{1}{2}\|\,|u| |\nabla u|\,\|^2_{L^2(Q_{\tau})}+CC_{\epsilon}^2
+C(\|u\|^2_{L^2(0,\tau;\operatorname{BMO}(\Omega))}+\epsilon)
\sup_{T^*-\epsilon<t< \tau}\|u(\cdot,t)\|^4_{L^4}.
\end{align*}
Since the above estimate holds for all $t$ with
$T^*-\epsilon<t<\tau$, we obtain
\begin{align*}
&\sup_{T^*-\epsilon<t< \tau}\|u(\cdot,t)\|^4_{L^4}
+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|\nabla u|^2|u|^2|\,dx\,dt
+\frac{1}{2}\int_{T^*-\epsilon}^{\tau}\int_{\Omega}\big|\nabla|u|^2\big|^2\,dx\,dt\\
&\leq \int_{\Omega}|u(\cdot, T^*-\epsilon)|^4 dx +CC^2_{\epsilon} \\
&\quad +C\Big(\|u\|^2_{L^2((T^*-\epsilon,\tau);\operatorname{BMO}(\Omega))}+\epsilon\Big)
\sup_{T^*-\epsilon<t< \tau}\|u(\cdot,t)\|^4_{L^4}.
\end{align*}
With sufficiently small $\epsilon$ so that
$\big(\|u\|^2_{L^2((T^*-\epsilon,\tau);\operatorname{BMO}(\Omega))}
+\epsilon\big) \leq \frac{1}{2C}$ with a constant $C>0$ in the above
estimate, we have
\begin{align*}
&\|u(\cdot,t)\|^4_{L_{x,t}^{4,\infty}(Q_{\tau})}
+\frac{1}{2}\|\,|\nabla u||u|\,\|^2_{L^2(Q_{\tau})}
+\frac{1}{2}\|\nabla|u|^2\|^2_{L^2(Q_{\tau})} \\
&\leq 2\|u(\cdot,T-\epsilon)\|^4_{L^4_{x}(\Omega)}+CC_{\epsilon}^2.
\end{align*}
For simplicity, we denote $Q_{\epsilon}=\Omega\times (T^*-\epsilon,
T^*)$. Since $\tau$ is arbitrary with $\tau<T^*$, we obtain
\begin{equation*}
\|u(\cdot,t)\|^4_{L_{x,t}^{4,\infty}(Q_{\epsilon})}
+\frac{1}{2}\big\||\nabla u||u|\big\|^2_{L^2(Q_{\epsilon})}
+\frac{1}{2}\|\nabla|u|^2\|^2_{L^2(Q_{\epsilon})}
\leq C,
\end{equation*}
where $C$ is a constant depending on
$\| u(\cdot,T^*-\epsilon)\|_{W^{1,2}(\Omega)}$. This is contrary
to the hypothesis of \eqref{1stpf-20}. Therefore, $T^*$ cannot be a
maximal time of existence less than or equal to $T$. This completes
the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
First, we consider the vorticity equation
\begin{equation}\label{vorticity}
 w_t-\Delta w+(u \cdot \nabla)w-(u \cdot \nabla)w=0.
\end{equation}
Multiplying the first equation of \eqref{vorticity} by  $w$,
integrating over $\Omega$, and adding them, we have
\begin{equation*}%\label{2ndpf-100}
\frac{1}{2}\frac{d}{dt}\int_{\Omega}|w|^2+\int_{\Omega}|\nabla w|^2
\leq\int_{\Omega} |w||\nabla u|
|w|+\int_{\partial\Omega} \big|\frac{\partial w}{\partial
n}\cdot w\big|:=II_1+II_2,
\end{equation*}
where we use Lemmas \ref{gradient} and \ref{estimate-boudnary}.
Using H\"{o}lder inequality and Lemma \ref{esti-bmo}, the term
$II_1$ is estimated as follows:
\[
II_1\leq \|\nabla u\|_{L^3(\Omega)}\|w\|^2_{L^3(\Omega)} \leq
C\|w\|^3_{L^3(\Omega)}\leq
C\|w\|^2_{L^2(\Omega)}\|w\|_{\operatorname{BMO}(\Omega)},
\]
Next, we can easily estimate $II_2$. Indeed, we use the Trace
theorem (see e.g., \cite[pp 257-258]{Evans}) and smoothness of
boundary to find
\begin{equation*}\label{est-ii2}
II_2\leq \int_{\partial\Omega} \big|\frac{\partial w}{\partial n}\cdot w\big|
\leq C\int_{\Omega}|w|^2,
\end{equation*}
Summing  the estimates $II_1$ and $II_2$, we obtain
\begin{equation}\label{sum}
\frac{d}{dt}\|w\|_{L^2}^2 +\|\nabla w\|_{L^2}^2\leq
C(1+\|w\|_{\operatorname{BMO}(\Omega)})\|w\|_{L^2}^2.
\end{equation}
Applying the Gronwall's inequality to \eqref{sum},
\begin{equation*}
\sup_{0<t<T}\|w(t)\|_{L^2}^2 +\int_0^T\|\nabla w\|_{L^2}^2\leq
C\|w_0\|_{L^2}^2,
\end{equation*}
which is the desired result.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm3}]
First, we note  that, without loss of generality, the mean value of
the pressure $\pi$ is assumed to be zero, namely $\int_{\Omega}
\pi(\cdot,t)dx=0$ for each time $t\in [0,T)$. We get $\pi$ satisfies
\begin{equation*}\label{poincare-estimate}
\|\pi\|_{L^2(\Omega)}\leq C\|\nabla \pi\|_{L^2(\Omega)}.
\end{equation*}
The proof of Theorem \ref{thm3} is similar to that of Theorem
\ref{thm1}. Indeed, from \eqref{fluid-inme-space}, we note that
\begin{align*}
&\frac{1}{4}\int_{\Omega}|u(\cdot,\tau)|^4 dx
 -\frac{1}{4}\int_{\Omega}|u(\cdot, T^*-\epsilon)|^4 dx \\
&+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|\nabla u|^2|u|^2 \,dx\,dt
+\frac{1}{2}\int_{T^*-\epsilon}^{\tau}\int_{\Omega}\big|\nabla|u|^2\big|^2\,dx\,dt\\
&\leq \int_{T^*-\epsilon}^{\tau}\int_{\Omega} |\pi|\, |u|\,|u||\nabla u|\,\,dx\,dt
+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|u|^3|\nabla u|\,dx\,dt:=III_1+III_2.
\end{align*}
Using H\"older's inequality, the first term $III_1$ can be estimated
as
\begin{align*}
III_1
&\leq \int_{T^*-\epsilon}^{\tau}\|\pi\|_{L^4}
\|u\|_{L^4}\big\||u||\nabla u|\big\|_{L^2}
 \leq C\int_{T^*-\epsilon}^{\tau} \|\pi\|_{L^4}
\|u\|_{L^4}\big\||u||\nabla u|\big\|_{L^2} \\
&\leq \int_{T^*-\epsilon}^{\tau}C\|\pi\|^{1/2}_{L^2}\|\pi\|^{1/2}_{\rm BMO}
\|u\|_{L^4}\big\||u||\nabla u|\big\|_{L^2} \\
&\leq C\|\nabla \pi\|^{1/2}_{L^2(Q_\tau)}
 \Big(\int_{T^*-\epsilon}^{\tau}\|\pi\|^2_{\rm BMO}
\|u\|^4_{L^4}\,dt\Big)^{1/4}\big\||u||\nabla u|\big\|_{L^2(Q_\tau)}
\end{align*}
For convenience of computations, we denote
$\mathcal{C}_{\epsilon}:=\|u(\cdot,T^*-\epsilon)\|_{W^{1,2}(\Omega)}$.
Using the estimate \eqref{stokes-estimate}, we obtain
\begin{equation}\label{estimate-2000}
\begin{aligned}
III_1
&\leq C\Big(\|(u \cdot \nabla) u\|^{1/2}_{L^2(Q_{\tau})}+\mathcal{C}_{\epsilon}\Big)
\Big(\int_{T^*-\epsilon}^{\tau}\|\pi\|^2_{\rm BMO}
\|u\|^4_{L^4}\,dt\Big)^{1/4} \big\||u||\nabla u|\big\|_{L^2(Q_\tau)}, \\
& \leq C\|(u \cdot \nabla) u\|^{1/2}_{L^2(Q_{\tau})}
 \Big(\int_{T^*-\epsilon}^{\tau}\|\pi\|^2_{\rm BMO}
\|u\|^4_{L^4}\,dt\Big)^{1/4} \big\||u||\nabla u|\big\|_{L^2(Q_\tau)}\\
&\quad +C\mathcal{C}_{\epsilon}\Big(\int_{T^*-\epsilon}^{\tau}\|\pi\|^2_{\rm BMO}
\|u\|^4_{L^4}\,dt\Big)^{1/4} \big\||u||\nabla u|\big\|_{L^2(Q_\tau)}.
\end{aligned}
\end{equation}
Following similar computations as in $I_2$, we obtain
\begin{equation}\label{estimate-2500}
III_2\leq C\big||u||\nabla u|\big|_{L^2(Q_{\tau})}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4}.
\end{equation}
Summing  \eqref{estimate-2000}-\eqref{estimate-2500} and using
Young's inequality, we obtain
\begin{align*}
&\frac{1}{4}\int_{\Omega}|u(\cdot,\tau)|^4 dx
 -\frac{1}{4}\int_{\Omega}|u(\cdot, T^*-\epsilon)|^4 dx \\
&+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|\nabla u|^2|u|^2 \,dx\,dt
+\frac{1}{2}\int_{T^*-\epsilon}^{\tau}\int_{\Omega}\big|\nabla|u|^2\big|^2\,dx\,dt\\
& \leq  C\|(u \cdot \nabla) u\|^{1/2}_{L^2(Q_{\tau})}
 \Big(\int_{T^*-\epsilon}^{\tau}\|\pi\|^2_{\rm BMO}
\|u\|^4_{L^4}\,dt\Big)^{1/4} \big\||u||\nabla u|\big\|_{L^2(Q_\tau)} \\
&\quad +C\mathcal{C}_{\epsilon}\Big(\int_{T^*-\epsilon}^{\tau}\|\pi\|^2_{\rm BMO}
\|u\|^4_{L^4}\,dt\Big)^{1/4} \big\||u||\nabla u|\big\|_{L^2(Q_\tau)} \\
&\quad +C\epsilon^\frac{1}{2}\big||u||\nabla u|\big|_{L^2(Q_{\tau})}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4} \\
&\leq \frac{1}{2}\big\||u||\nabla u|\big\|^2_{L^2(Q_{\tau})}+CC_{\epsilon}^4
+C\Big[\int_{T^*-\epsilon}^\tau\|\pi(\cdot,t)\|^2_{\rm BMO}\,dt+\epsilon\Big]
\sup_{T^*-\epsilon<t< \tau}\|u(\cdot,t)\|^4_{L^4}.
\end{align*}
With sufficiently small $\epsilon$ so that
\[
\Big(\int_{T^*-\epsilon}^\tau\|\pi(\cdot,t)\|^2_{\rm BMO}\,dt+\epsilon\Big)
\leq \frac{1}{2C}
\]
 with a constant $C$ in the above estimate, we have
\begin{align*}
&\|u(\cdot,t)\|^4_{L_{x,t}^{4,\infty}(Q_{\tau})}
+\frac{1}{2}\|\,|\nabla u||u|\|^2_{L^2(Q_{\tau})}
+\frac{1}{2}\,\|\nabla|u|^2\|^2_{L^2(Q_{\tau})} \\
&\leq 2\|u(\cdot,T-\epsilon)\|^4_{L^4_{x}(\Omega)}+CC_{\epsilon}^4.
\end{align*}
By the same argument as in the proof of Theorem \ref{thm1}, we
finally obtain
\begin{equation*}
\|u(\cdot,t)\|^4_{L_{x,t}^{4,\infty}(Q_{\epsilon})}
+\frac{1}{2}\big\||\nabla u||u|\big\|^2_{L^2(Q_{\epsilon})}
+\frac{1}{2}\|\nabla|u|^2\|^2_{L^2(Q_{\epsilon})}
\leq C,
\end{equation*}
where $C$ is a constant depending on
$\|u(\cdot,T^*-\epsilon)\|_{W^{1,2}(\Omega)}$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm4}]
This  proof  is similar to that of Theorem \ref{thm1}.
 Indeed, from From \eqref{fluid-inme-space}, we note that
\begin{align*}
&\frac{1}{4}\int_{\Omega}|u(\cdot,\tau)|^4 dx
 -\frac{1}{4}\int_{\Omega}|u(\cdot, T^*-\epsilon)|^4 dx \\
&+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|\nabla u|^2|u|^2 \,dx\,dt
+\frac{1}{2}\int_{T^*-\epsilon}^{\tau}
\int_{\Omega}\big|\nabla|u|^2\big|^2\,dx\,dt\\
&\leq \int_{T^*-\epsilon}^{\tau}\int_{\Omega} |\nabla \pi|\, |u|^2 |u|\,dx\,dt
+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|u|^3|\nabla u|\,dx\,dt:=IV_1+IV_2.
\end{align*}
 Using H\"older's inequality, the first term $IV_1$ can be
estimated as
\begin{align*}
IV_1
&\leq \int_{T^*-\epsilon}^{\tau}\|\nabla\pi\|_{L^4}
\|u\|^3_{L^4} \leq C\int_{T^*-\epsilon}^{\tau}
\|\nabla\pi\|^{1/2}_{L^2}\|\nabla\pi\|^{1/2}_{\rm BMO}\|u\|^3_{L^4} \\
&\leq C\|\nabla \pi\|^{1/2}_{L^2(Q_{\tau})}
\Big(\int_{T^*-\epsilon}^\tau\|\nabla
\pi(\cdot,t)\|^{2/3}_{\rm BMO}\,dt\Big)^{3/4}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^3_{L^4},
\end{align*}
For convenience of computations, we denote
$\mathcal{C}_{\epsilon}:=\|u(\cdot,T^*-\epsilon)\|_{W^{1,2}(\Omega)}$.
Using the estimate \eqref{stokes-estimate}, we obtain
\begin{equation}\label{estimate-2000b}
\begin{aligned}
IV_1
&\leq C\Big(\|(u \cdot \nabla) u\|^{1/2}_{L^2(Q_{\tau})}
 +\mathcal{C}_{\epsilon}\Big)\Big(\int_{T^*-\epsilon}^\tau\|\nabla
\pi(\cdot,t)\|^{2/3}_{\rm BMO}\,dt\Big)^{3/4} \\
&\quad\times \sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^3_{L^4}, \\
&\leq C\|(u \cdot \nabla) u\|^{1/2}_{L^2(Q_{\tau})}
 \Big(\int_{T^*-\epsilon}^\tau\|\nabla
\pi(\cdot,t)\|^{2/3}_{\rm BMO}\,dt\Big)^{3/4}
 \sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^3_{L^4} \\
&\quad +C\mathcal{C}_{\epsilon}\Big(\int_{T^*-\epsilon}^\tau\|\nabla
\pi(\cdot,t)\|^{2/3}_{\rm BMO}\,dt\Big)^{3/4}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^3_{L^4}.
\end{aligned}
\end{equation}
Following similar computations as in $I_2$, we obtain
\begin{equation}\label{estimate-2500b}
IV_2\leq C\epsilon^{1/2}\big||u||\nabla u|\big|_{L^2(Q_{\tau})}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4}.
\end{equation}
Summing \eqref{estimate-2000}-\eqref{estimate-2500} and using
Young's inequality, we obtain
\begin{align*}
&\frac{1}{4}\int_{\Omega}|u(\cdot,\tau)|^4 dx
-\frac{1}{4}\int_{\Omega}|u(\cdot, T^*-\epsilon)|^4 dx \\
&+\int_{T^*-\epsilon}^{\tau}\int_{\Omega}|\nabla u|^2|u|^2 \,dx\,dt
+\frac{1}{2}\int_{T^*-\epsilon}^{\tau}\int_{\Omega}\big|\nabla|u|^2\big|^2\,dx\,dt\\
&\leq C\|(u \cdot \nabla) u\|^{1/2}_{L^2(Q_{\tau})}
\Big(\int_{T^*-\epsilon}^\tau\|\nabla \pi(\cdot,t)\|^{2/3}_{\rm BMO}\,dt\Big)^{3/4}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^3_{L^4} \\
&\quad +C\mathcal{C}_{\epsilon}\Big(\int_{T^*-\epsilon}^\tau
 \|\nabla \pi(\cdot,t)\|^{2/3}_{\rm BMO}\,dt\Big)^{3/4}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^3_{L^4} \\
&\quad +C\epsilon^{1/2}\big||u||\nabla u|\big|_{L^2(Q_{\tau})}
\sup_{T^*-\epsilon<t<\tau}\|u(\cdot,t)\|^2_{L^4} \\
& \leq \frac{1}{2}\|\,|u| |\nabla u|\,\|^2_{L^2(Q_{\tau})}+CC_{\epsilon}^4
+C\Big(\int_{T^*-\epsilon}^\tau\|\nabla \pi(\cdot,t)\|^{2/3}_{\rm BMO}\,dt
 +\epsilon\Big) \\
&\quad\times \sup_{T^*-\epsilon<t< \tau}\|u(\cdot,t)\|^4_{L^4}.
\end{align*}
With sufficiently small $\epsilon$ so that
\[
\Big(\int_{T^*-\epsilon}^\tau\|\nabla
\pi(\cdot,t)\|^{2/3}_{\rm BMO}\,dt+\epsilon \Big) \leq \frac{1}{2C}
\]
with a constant $C$ in the above estimate, we have
\begin{align*}
&\|u(\cdot,t)\|^4_{L_{x,t}^{4,\infty}(Q_{\tau})}
+\frac{1}{2}\|\,|\nabla u||u|\,\|^2_{L^2(Q_{\tau})}
+\frac{1}{2}\|\nabla|u|^2\|^2_{L^2(Q_{\tau})} \\
&\leq 2\|u(\cdot,T-\epsilon)\|^4_{L^4_{x}(\Omega)}+CC_{\epsilon}^4.
\end{align*}
By the same argument from the proof of Theorem \ref{thm1}, we
finally obtain the desired result.
\end{proof}

\begin{remark} \rm
The arguments of Theorems \ref{thm1}--\ref{thm3} also hold for a
whole space $\mathbb{R}^n$ because Lemma \ref{lem1} also established for
these cases.
\end{remark}


\subsection*{Acknowledgments}
I would like to thank the anonymous referee for the useful comments.
Jae-Myoung Kim was supported by grants  NRF-20151009350 and
NRF-2016R1D1A1B03930422.

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\end{document}